VNU JOURNAL OF SCIENCE Mathematics - Physics t.XVIII n°l - 2002 ON TH E LIN EA R RADICAL OF LATTICES N guyen D ue D at F n cu liy ' M ỉìLlicm nìics, College o f N n tui’fil sricnces V N Ư IỈ I n t r o d u c t io n T lir radical theory is an im portant tool for stud ying th e structure and th e cliissiliration of algebraic structures It attra cts large interest of many authors iỉ r> rii besides T there exist linear congruencies, Nam ely, Pi consists o f the classes {a h \ %{() r} and p consists o f the classes {0, a, />}, {c, 1} T hus ( iV.5 ) = /?! n />'2 which consists of classes = { a 6} {(;}, {0 }, {1 } 2.6 Proposition ( Ì ) For ỈÌÌÌ Hi bitrnrv Ini tier L %the quotient lattice L / r ( L ) is distributive (2) L('t D h r ri latticc Then D is (listrihut i w if find only ifÉr ( D ) — A Proof (I ) We have r ( L ) = n {/),ị/ f I } where /),./ / are linear congruencies on /.As well-known L / r ị L ) i.s tlu* Mil>-h tu those of radicals o f rings Application 111 th is section wo present an application o f the linear radical to classifying modular lattices Here, for lattices, it is particular that p is r-sem i sim ple class Therefore, the classification problem is o f interest itself For exam ple, consider a m odular lattice A / A s in E xam ple 2.5 we see that M is r-iadieal lattice On the other hand, the sublattices of M which are isomorphic to A/.J prevent M from being d istribu tive Since M / r ( M ) is distributive., each sublattice isomorphic to M\\ belongs to on e of equivalence classes of r ( M ) Based upon t.ỈK' above reasons we arrive at th e stu d y of the particular modular bit tiers form ulated in the following th eorem T h e o r e m Lot M !)(' a m o d u la r lã t t k r which is not distributive Let ÌH' ỈÌ sublattice o f M such tlmt: 1) A is convex 2) A is r-nulicnl 3) A contains all sublattices o f M which are isom orphic to A/3 Then r ( M ) has one class equal to A, the other classes ( i f they exist) are distributive Proof We can assum e that M A and use th e following lemma L e m m a I f b G M A then in M there exists a two-classes congruence, OJIO class o f which contains A , the other contains b Proof For I) and A we have the following alternatives: (I) 3a € A (I < b (II) E ither Va € A a\\b or 3a € A a > b For case (I), consider the principal filter generated by I) : F(b) = {./• € M \.i > b) If 3c € F ( b ) n A then (I < I) < (\ it im plies th at b € A (due to convexity o f 4) but it contradicts the assum ption T hus F { b ) n A = We den ote by T the' ffiinily of all filters containing b lmt not any element of A Consider T w ith relation c Let { / \ | i € / } be a chain on T it is easy to deduce* that N guyen D uc D at \ F , / i /} € T Due to Zorn's Lemma, there exists a maximal rlriiu nt of T which we denote 1)V F Wo consider the ideal generated by A again /(^4) = {;/• M/.V (I for som e a *4} Obviously ( i ) ) n F = For case (II), we take th e principal ideal generated by b : J(.-l) {./.* G M \ x < b} Obviously, J ( b ) n A = Sim ilarly to case (I) we can deduce that there exist the maximal ideal / and the maxim al filter F such th at I) € »/,.7 / = and ỗ F, f F = Now for both cases (I) and (II) we shall prove that / u F —.Ì/ w v suppose that 3c* € A /, c Ệ J u F First we prove the assertion: (i) \ j £ J J V c £ F hulivd if V / €